Problem: Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$  Find the maximum value of $xy.$
By AM-GM,
\[60 = 4x + 9y \ge 2 \sqrt{(4x)(9y)} = 2 \sqrt{36xy} = 12 \sqrt{xy},\]so $\sqrt{xy} \le 5.$  Hence, $xy \le 25.$

Equality occurs when $4x = 9y.$  Along with the condition $4x + 9y = 60,$ we can solve to get $x = \frac{15}{2}$ and $y = \frac{10}{3},$ so the maximum value is $\boxed{25}.$